TRAINS OF REASONING. 147 



spend regulai-ly to variations of quantity either in those same or in 

 some other phenomena ; every formula of mathematics applicable to 

 quantities which vary in that paiticular manner, becomes a mark of a 

 corresponding general truth respecting the variations in quality which 

 accompany them : and the science of quantity being (as far as any 

 science can be) altogether deductive, the theory, of that particular kind 

 of qualities becomes, to this extent, deductive likewise. 



The most striking instance in poiut which history affords, (though 

 not an example of an experimental science rendered deductive, but of 

 an unparalleled extension given to the deductive process in a science 

 which was deductive already,) is the revolution in geometry which 

 originated with the illustrious Descartes, and was completed by Clai- 

 raut. These philosophers remarked, that to every variety of position 

 in points, direction in lines, or form in curves or surfaces, (all of which 

 are Qualities,) there corresponds a peculiar relation of quantity between 

 either two or three rectilineal coordinates ; insomuch that if the law 

 were known according to which those coordinates vary relatively to 

 one another, every other geometrical property of the line or surface in 

 question, whether relating to quantity or quality, would be capable of 

 being infen-ed. Hence it followed that every geometrical question 

 could be solved, if the corresponding algebraic^al one could ; and 

 geometry received an accession (actual or potential) of new tiTiths, cor- 

 responding to every property of numbers which the prf)gress of the 

 calculus had brought, or might in future bring, to light. In the same 

 general manner, mechanics, astronomy, and in a less degree, every 

 branch of natural philosophy commonly so called, have been made 

 algebraical. The varieties of physical phenomena with which those 

 sciences are conversant, have been found to answer to determinable 

 varieties in the quantity of some circumstance or other ; or at least to 

 varieties of form or position, for which corresponding equations of 

 quantity had already been, or were susceptible of being, discovered 

 by geometers. 



In these various transformations, 'the propositions of the science of 

 number do but fulfil the function proper to all propositions forming a 

 train of reasoning, viz., that of enabUng us to arrive in an indirect 

 method, by marks of marksj, at such of the propeities of objects as we 

 cannot directly ascertain (or not so conveniently) by experiment. 

 We travel from a given visible or tangible fact, throuo-h the truths 

 of numbers, to the fact sought. The given fact is a mark that a cer- 

 tain relation subsists between the quantities of some of the elements 

 concerned ; while the fact sought presupposes a ceitain relation 

 between the quantities of some other elements : now, if these last 

 quantities are dependent in some known manner upon the former, or 

 vice versa, we can argue from the numeiical relation between the one 

 set of quantities, to detennine that which subsists between the other 

 set ; the theorems of the calculus affording the intermediate links. 

 And thus the one of the two physical facts becomes a mark of the 

 other, by being a mark of a mark of a mark of it. 



